\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 77, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/77\hfil Sign-changing solutions] {Sign-changing solutions of a fourth-order elliptic equation with supercritical exponent} \author[K. Ould Bouh \hfil EJDE-2014/77\hfilneg] {Kamal Ould Bouh} % in alphabetical order \address{Kamal Ould Bouh \newline Department of Mathematics, Taibah University, P.O. Box: 30002, Almadinah Almunawwarah, Saudi Arabia} \email{hbouh@taibahu.edu.sa, kamal\_bouh@yahoo.fr} \thanks{Submitted November 15, 2013. Published March 19, 2014.} \subjclass[2000]{35J20, 35J60} \keywords{Sign-changing solutions; critical exponent; bubble-tower solution} \begin{abstract} In this article we study the nonlinear elliptic problem involving nearly critical exponent \begin{gather*} \Delta^2 u = |u|^{8/(n-4)+\varepsilon}u\quad\text{in } \Omega, \\ \Delta u=u = 0\quad \text{on } \partial \Omega, \end{gather*} where $\Omega $ is a smooth bounded domain in $\mathbb{R}^n $ with $n \geq 5 $, and $\varepsilon$ is a positive parameter. We show that, for $\varepsilon$ small, there is no sign-changing solution with low energy which blow up at exactly two points. Moreover, we prove that this problem has no bubble-tower sign-changing solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of results} In this article, we consider the semi-linear elliptic problem with supercritical nonlinearity \begin{equation} \label{ePe} \begin{gathered} \Delta^2 u= |u|^{p-1+\varepsilon}u \quad \text{in } \Omega, \\ \Delta u= u= 0 \quad \text{on } \partial \Omega, \end{gathered} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\geq 5$, $\varepsilon$ is a positive real parameter and $p+1= \frac{2n}{n-4}$ is the critical Sobolev exponent for the embedding of $H^2(\Omega)\cap H_0^1(\Omega)$ into $L^{p+1}(\Omega)$. Problem \eqref{ePe} is related to the limiting problem (when $\varepsilon=0$) which exhibits a lack of compactness. In fact, van Der Vorst \cite{V1, V2} (see also \cite{Mi}) showed that \eqref{ePe} with $\varepsilon=0$ has no positive solutions if $\Omega$ is a starshaped domain. Whereas Ebobisse and Ould Ahmedou \cite{EA} proved that \eqref{ePe} with $\varepsilon=0$ has a positive solution provided that some homology group of $\Omega$ is non trivial. This topological condition is sufficient, but not necessary, as examples of contractible domains $\Omega$ on which a positive solution exists as shown in \cite{GGS} (see also \cite{GGSw}). Note that some problems of type \eqref{ePe} were studied in case of Riemannian manifolds, see for example \cite{HRW} and \cite{PV}. In view of this qualitative change of the situation for \eqref{ePe} with $\varepsilon=0$, it is interesting to study the problem \eqref{ePe} with $\varepsilon<0$ and $\varepsilon>0$ and to understand what happens to the solutions of \eqref{ePe} (\text{if }they exist) as $\varepsilon \to 0$. Observe that, when $\varepsilon < 0$, the existence of solutions of \eqref{ePe} has been proved in \cite{BE,BG} (see \cite{Bar,BarW,CCN} for the Laplacian case) for each $\varepsilon \in (1-p, 0)$. For the positive solutions, Chou and Geng \cite{CG} made the first study, when $\Omega$ is a convex domain. They gave the asymptotic behavior of the low energy positive solution. They used the method of moving planes to show that the blow-up point is away from the boundary of the domain. The process is standard if the domain is convex. We note that, for non convex regions, this method still works in the Laplacian case through the applications of Kelvin transformations, see \cite{H} (since the problem is invariant under these transformations). However, the Navier boundary conditions are not invariant under the Kelvin transformation of the biharmonic operator. But the method of moving planes also works for convex domains, see \cite{CG}. To remove the convexity assumption, Ben Ayed and El Mehdi \cite{BE} used another method based on some ideas introduced by Bahri in \cite{B1}. This result is the analogous one to the one in \cite{R2} and \cite{H} where the Laplacian operator was studied. Concerning the supercritical case, $\varepsilon>0$, the problem \eqref{ePe} becomes more delicate since we lose the Sobolev embedding which is an important point to overcome. We recall that, when the biharmonic operator in \eqref{ePe} is replaced by the Laplacian one, there are many works devoted to the study of the positive solutions of the counterpart of \eqref{ePe}. It was proved in \cite{BEGR} that \eqref{ePe} has no positive solution which blows up at a single point. This result shows that the situation is different from the subcritical case. However, Del Pino et al \cite{DFM} (see also \cite{KR}) gave an existence result for two blow up points, provided that $\Omega$ satisfies some geometrical conditions. In sharp contrast to this, very little study has been made concerning the sign-changing solutions, see \cite{BB1}. It is well known that problem \eqref{ePe} (with $\varepsilon<0$) has always a positive least energy solution $u_\varepsilon$ which is obtained by solving the variational problem $$ \inf J(u) \quad \text{where } J(u):= \frac{\int _\Omega |\Delta u|^2}{\big(\int _\Omega |u|^{p+1+\varepsilon}\big)^{2/(p+1+\varepsilon)}} \quad u\in H^2(\Omega)\cap H^1_0(\Omega), \quad u\not\equiv 0. $$ Removing the assumption of the positivity of the solutions, the study of the asymptotic behavior becomes difficult. The main difficulty is that the limit problem, after a change of variable, which is \begin{equation}\label{i:1} \Delta^2 u= |u|^{p-1}u \quad \text{in } \mathbb{R}^n, \end{equation} has many sign-changing solutions which are unknown. However, an interesting information about the energy shows that \cite[Lemma 2]{GGS} \begin{equation}\label{i:2} \int_{\mathbb{R}^n} |\Delta w|^2 > 2S^{n/4}, \end{equation} for each sign-changing solution $w$ of \eqref{i:1}, where $S$ denotes the best minimizers of the Sobolev inequality on the whole space, that is \begin{equation*} S =\inf\{|\Delta u|^{2}_{L^2(\mathbb{R}^n)}|u|^{-2}_{L^{{2n}/(n-4)}(\mathbb{R}^n)} : \Delta u\in L^2 ,u\in L^{2n/(n-4)} ,u\not\equiv 0 \}. \end{equation*} When we add the positivity assumption, the solutions of \eqref{i:1} are the family \begin{equation}\label{i:3} \delta _ {(a,\lambda )}(x) = c_0 \frac {\lambda ^{(n-4)/2}}{(1+\lambda^2|x-a|^2)^{(n-4)/2}},\quad c_0=\big(n(n-4)(n^2-4)\big)^{(n-4)/8}, \end{equation} with $\lambda >0$ and $a \in \mathbb{R}^n$. The space $H^2(\Omega)\cap H_0^1(\Omega)$ is equipped with the norm $\|\cdot\|$ and its corresponding inner product $\langle\cdot,\cdot\rangle$ defined by \begin{equation} \|u\|=\Big(\int_\Omega |\Delta u|^2\Big)^{1/2}, \quad \langle u,v\rangle=\int_\Omega \Delta u \Delta v, \quad u, v \in H^2(\Omega)\cap H_0^1(\Omega). \end{equation} When we study problem \eqref{i:1} in a bounded smooth domain $\Omega$, we need to introduce the function $P\delta _ {(a,\lambda)}$ which is the projection of $\delta _ {(a,\lambda)}$ on $H^1_0(\Omega)$. It satisfies \begin{equation*} \Delta^2 P\delta _{(a, \lambda)} = \Delta^2\delta _{(a,\lambda)} \quad \text{in } \Omega ; \quad \Delta P\delta _{(a, \lambda)}=P\delta _{(a, \lambda)}= 0 \quad \text{on } \partial \Omega. \end{equation*} These functions are almost positive solutions of \eqref{ePe}. Our first result deals with the low energy sign-changing solution of \eqref{ePe} with $\varepsilon >0$. We prove that there is no solution which blows up at exactly two points. More precisely, we have the following result. \begin{theorem}\label{t:11} Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$. There exists $\varepsilon_0>0$, such that for each $\varepsilon \in (0,\varepsilon_0)$, problem \eqref{ePe} has no sign-changing solution $u_\varepsilon$ which satisfies \begin{align}\label{n:1} u_\varepsilon= P\delta_{(a_{\varepsilon,1},\lambda_{\varepsilon,1})} -P\delta_{(a_{\varepsilon,2},\lambda_{\varepsilon,2})}+v_\varepsilon, \end{align} with the $L^{\infty}$-norm of $u_\varepsilon$ at the power $\varepsilon$ ($|u_\varepsilon|_{\infty}^{\varepsilon}$) begin bounded and \begin{gather*} a_{\varepsilon,i}\in \Omega,\quad \lambda_{\varepsilon,i}d(a_{\varepsilon,i}, \partial \Omega) \to \infty \quad\text{for } i=1,2 \\ \langle P\delta_{(a_{\varepsilon,1},\lambda_{\varepsilon,1})}, P\delta_{(a_{\varepsilon,2},\lambda_{\varepsilon,2})}\rangle \to 0 \quad \text{and}\quad \|v_\varepsilon\|\to 0 \quad \text{as } \varepsilon \to 0 . \end{gather*} \end{theorem} We point out that there are other important phenomena in sign-changing solutions. Indeed, it is possible to find solutions having bubble over bubble (bubble-tower solutions). In the case of the Laplacian operator, Pistoia and Weth \cite{PW} constructed a family of sign-changing solutions of \eqref{ePe} ($\varepsilon<0$) with $k$ bubbles, $k\geq2$, concentrated at the same point. This result gives a new phenomenon compared with the positive case. In their paper, they conjectured that this phenomenon cannot appear when $\varepsilon>0$. In \cite{BB1}, we gave an affirmative answer for the conjecture of Pistoia and Weth. The following result deals with phenomenon of bubble-tower solutions for the biharmonic problem \eqref{ePe} with supercritical exponent. \begin{theorem}\label{t:14} Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$. There exists $\varepsilon_0>0$, such that for each $\varepsilon \in (0,\varepsilon_0)$, problem \eqref{ePe} has no solution $u_\varepsilon$ of the form \begin{equation} \label{i:4} u_\varepsilon = \sum _{i=1}^k (-1)^{i+1}P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})} + v_\varepsilon, \end{equation} with $ \lambda_{\varepsilon ,1}\leq \lambda_{\varepsilon ,2}\leq \dots \leq \lambda_{\varepsilon ,k}$ and $|u_\varepsilon|_{\infty}^{\varepsilon}$ bounded, where $k\geq2$, $a_{\varepsilon ,i}\in\Omega$, $\min(\lambda _{\varepsilon ,i},\lambda _{\varepsilon,j}) | a_{\varepsilon ,i}-a_{\varepsilon ,j}|$ is bounded, and $v_\varepsilon \to 0$ in $H^1_0(\Omega)$, $\lambda _{\varepsilon,i}d(a_{\varepsilon ,i},\partial \Omega) \to +\infty$, $\langle P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})},P\delta _{(a_{\varepsilon ,j}, \lambda _{\varepsilon ,j})}\rangle \to 0$, for $i\neq j$, as $\varepsilon \to 0$. \end{theorem} Note that Theorem \ref{t:14} deals with the bubble-tower solutions at one point. However Theorem \ref{t:11} says that there are no solutions which blow up at two points. Combining the ideas of the proof of Theorems \ref{t:11} and \ref{t:14}, we are able to prove the following result. \begin{theorem}\label{t:17} Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$. There exists $\varepsilon_0>0$, such that for each $\varepsilon \in (0,\varepsilon_0)$, problem \eqref{ePe} has no solution $u_\varepsilon$ of the form \begin{eqnarray}\label{ii:4} u_\varepsilon = \sum _{i=1}^m (-1)^{i+1}P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})} + \sum _{i=m+1}^k (-1)^{i-m}P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})} + v_\varepsilon:= u_\varepsilon^1+u_\varepsilon^2+v_\varepsilon, \end{eqnarray} with $ |u_\varepsilon|_{\infty}^{\varepsilon}$ bounded, $\|v_\varepsilon\| \to 0$, $a_{\varepsilon ,i}\to a$ for each $i\leq m$, $a_{\varepsilon ,i} \to b$ for each $i\geq m+1$ with $a\ne b$, and, for $j=1,2$, if $u_\varepsilon^j$ contains more than one bubble then it satisfies the assumptions of Theorem \ref{t:14}. \end{theorem} The proof of our results will be by contradiction. Thus, throughout this paper we will assume that there exist solutions $(u_\varepsilon)$ of \eqref{ePe} which satisfy \eqref{n:1} or \eqref{i:4}. In Section 2, we will obtain some information about such $(u_\varepsilon)$ which allow us to develop Sections 3 which deal with some useful estimates to the proof of our Theorems. Finally, in Section 4, we combine these estimates to obtain a contradiction. Hence the proof of our results. \section{Preliminary results} In this Section, we assume that there exist solutions $(u_\varepsilon)$ of \eqref{ePe} which satisfy \begin{equation}\label{e:20} u_\varepsilon = \sum _{i=1}^k (-1)^{i+1}P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})} + v_\varepsilon, \end{equation} with $|u_\varepsilon|_{\infty}^{\varepsilon}$ bounded, $k\geq 2$, $a_{\varepsilon,i}\in\Omega$, and as $\varepsilon \to 0$, $\|v_\varepsilon\| \to 0$, $\lambda_{\varepsilon,i}d(a_{\varepsilon ,i},\partial \Omega) \to +\infty$, $\langle P\delta_{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})}, P\delta _{(a_{\varepsilon ,j}, \lambda _{\varepsilon ,j})}\rangle \to 0$ for $i\neq j$. We will collect some useful information used in the next sections. First, from \eqref{e:20}, it is easy to see that the following remark holds. \begin{remark}\label{l:21} \rm Let $(u_\varepsilon)$ be a family of sign-changing solutions of \eqref{ePe} satisfying \eqref{e:20}. Then \begin{itemize} \item[(i)] $ u_\varepsilon \rightharpoonup 0$ as $\varepsilon \to 0$, \item[(ii)] $\int_{\Omega}|u_\varepsilon|^{p+1+\varepsilon} =\int_{\Omega}|\Delta u_\varepsilon|^2=kS^{n/4}+o(1)$, \item[(iii)] $M_{\varepsilon,+}:=\max_\Omega u_\varepsilon \to +\infty$, $M_{\varepsilon,-}:=-\min_\Omega u_\varepsilon \to +\infty$ as $\varepsilon \to 0$. \end{itemize} \end{remark} Secondly, arguing as in \cite{BC} and \cite{R}, we see that for $u_\varepsilon$ satisfying \eqref{e:20}, there is a unique way to choose $\alpha_i$, $a_i$, $\lambda _i$ and $v$ such that \begin{equation}\label{e:11} u_\varepsilon = \sum_{i=1}^k (-1)^{i+1}\alpha _iP\delta _{(a_i, \lambda _i)} + v, \end{equation} with \begin{equation} \label{e:12} \begin{gathered} \alpha_i \in \mathbb{R}, \quad \alpha_i \to 1,\\ a_i \in \Omega, \quad \lambda_i \in \mathbb{R}^*_+,\quad \lambda_i d(a_i , \partial\Omega )\to +\infty,\\ v \to 0 \quad\text{in } H^2(\Omega)\cap H^1_0(\Omega), \quad v \in E, \end{gathered} \end{equation} where $E$ denotes the subspace of $H^1_0(\Omega)$ defined by \begin{equation}\label{E} E:=\big\{w: \langle w,\varphi\rangle=0,\; \forall\varphi\in \operatorname{span}\{P\delta _i, \partial P\delta_i/\partial \lambda_i, \partial P\delta _i/\partial a^j_i,\, i\leq k; j\leq n \}\big\}. \end{equation} Here, $a_i^j$ denotes the $j$-th component of $a_i$ and in the sequel, in order to simplify the notations, we set \begin{equation} \delta _{(a_i , \lambda _i) }=\delta _i, \quad P\delta _{(a_i , \lambda _i) }=P\delta _i. \end{equation} In the following, we always assume that $u_\varepsilon$ (which satisfies \eqref{e:20}) is written as in \eqref{e:11} and \eqref{e:12} holds. \begin{lemma}\label{l:23} Let $u_\varepsilon$ satisfying the assumption of above theorems. Then $\lambda _i$ occurring in \eqref{e:11} satisfies \begin{equation}\label{e:28} \lambda _i ^\varepsilon \to 1 \quad \text{as $\varepsilon \to 0$, for each } i\leq k. \end{equation} \end{lemma} \begin{proof} By Remark \ref{l:21}, we know that \begin{equation}\label{e:29} \int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon} = kS^{n/4} + o(1) \quad \text{as } \varepsilon \to 0. \end{equation} Furthermore, \begin{equation}\label{e:210} \int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon} =\int_{\Omega} -\Delta u_\varepsilon u_\varepsilon = \int_{\Omega}|u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon \Big(\sum (-1)^{i+1}\alpha _iP\delta_i\Big)+O(\|v\|^2). \end{equation} Observe that \begin{equation} \label{e:211} \begin{aligned} &\int _{\Omega}|u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon \Big(\sum (-1)^{i+1}\alpha _iP\delta_i\Big) \\ &=\sum\alpha _i ^{p+\varepsilon +1}\int _{\Omega}P\delta _i^{p+\varepsilon +1} +O\Big(\sum_{j\neq i}\int _{\Omega} P\delta _i^{p+\varepsilon}P\delta _j\Big) \\ &\quad +O\Big(\sum\int _{\Omega}\alpha _i P\delta _i^{p+\varepsilon}|v|+\sum\int _{\Omega}\alpha _i P\delta _i |v|^{p+\varepsilon}\Big) :=\sum A_i, \end{aligned} \end{equation} where \begin{equation*} A_i:= \alpha _i ^{p+\varepsilon +1}\int _{\Omega}P\delta _i ^{p+\varepsilon +1}+O\Big(\sum_{j\neq i}\int _{\Omega} P\delta _i^{p+\varepsilon}P\delta _j+\int _{\Omega}P\delta _i^{p+\varepsilon}|v|+\int _{\Omega}P\delta _i|v|^{p+\varepsilon}\Big). \end{equation*} Easy computations show that \begin{gather*} \int _{\Omega}P\delta _i ^{p+1+\varepsilon}= \lambda _i ^{\varepsilon(n-4)/2}\left(S^{n/4} +o(1)\right),\\ \int _{\Omega}P\delta _i^{p+\varepsilon}|v|=\lambda _i ^{\varepsilon(n-4)/2}O(|v |_{L^{p+1}}),\\ \int _{\Omega}P\delta _i|v|^{p+\varepsilon}=\lambda _i ^{\varepsilon(n-4)/2}O(|v|_{L^{p+1}}^{p+\varepsilon}). \end{gather*} Recall that for $i\neq j$ (see \cite{B1}) \begin{equation*} \int _{\mathbb{R}^n} \delta _i^p\delta _j =\int _{\mathbb{R}^n} \delta _j^p\delta _i=c\varepsilon_{ij}+ O\Big(\varepsilon_{ij}^{n/(n-4)}\log\varepsilon_{ij}^{-1}\Big), \end{equation*} where $c$ is a positive constant and, for $i\neq j$, $\varepsilon_{ij}$ is defined by \begin{equation}\label{e:215} \varepsilon_{ij}=\Big(\frac{\lambda_i}{\lambda_j}+\frac{\lambda_j}{\lambda_i} +\lambda_i\lambda_j|a_i-a_j|^2\Big)^{(4-n)/2}. \end{equation} Hence, we obtain \begin{equation*} \int _{\Omega} P\delta _i^{p+\varepsilon}P\delta _j =O\Big(\lambda _i ^{\varepsilon(n-4)/2}\int _{\Omega} P\delta _i^pP\delta _j\Big) =\lambda _i ^{\varepsilon(n-4)/2}O(\varepsilon_{ij}), \quad \text{for } i\neq j. \end{equation*} Thus \begin{equation}\label{e:212} A_i= \alpha _i ^{p+\varepsilon +1}\lambda _i ^{\varepsilon(n-4)/2}\left(S^{n/4} + o(1)\right). \end{equation} Therefore \eqref{e:210}, \eqref{e:211}, and \eqref{e:212} provide us with \begin{equation}\label{e:213} \int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon} =\Big(\sum \alpha _i ^{p+1+\varepsilon} \lambda _i ^{\varepsilon(n-4)/2}\Big)\Big(S^{n/4}+o(1)\Big) +o(1). \end{equation} Combining \eqref{e:29}, \eqref{e:213}, and the fact that $\alpha _i$ satisfies \eqref{e:12}, the lemma follows. \end{proof} \begin{remark}[\cite{BEGR,R2} ]\label{r:24} \rm We recall the estimate \begin{equation}\label{e:214} \delta _i^\varepsilon (x) - c_0^\varepsilon \lambda _i ^{\varepsilon (n-4)/2} = O\left(\varepsilon \log (1+\lambda _i ^2 |x-a_i|^2)\right) \quad \text{in } \Omega , \end{equation} which will be very useful in the next section. \end{remark} \section{Some useful estimates} As usual in this type of problems, we first deal with the $v$-part of $u_\varepsilon$, in order to show that it is negligible with respect to the concentration phenomenon. \begin{lemma}\label{l:41} The function $v$ defined in \eqref{e:11}, satisfies the estimate \begin{align*} &\|v\| \\ &\leq c\varepsilon+ c \begin{cases} \sum_i \frac{1}{(\lambda_id_i)^{n-4}}+ \sum_{i\neq j}\varepsilon_{ij}(\log\varepsilon_{ij}^{-1})^{(n-4)/n} &\text{if } n < 12,\\[4pt] \sum_i \frac{1}{(\lambda_id_i)^{(n+4)/2-\varepsilon(n-4)}} +\sum_{i\neq j}\varepsilon_{ij}^{(n+4)/2(n-4)} (\log \varepsilon_{ij}^{-1})^{(n+4)/2n} &{if } n \geq 12, \end{cases} \end{align*} where $\varepsilon_{ij}$ is defined in \eqref{e:215} and $d_i:=d(a_i,\partial\Omega)$ for $i\leq k$. \end{lemma} \begin{proof} Since $u_\varepsilon = \sum (-1)^{i+1}\alpha _iP\delta _i+v$ is a solution of \eqref{ePe} and $v \in E$ (see \eqref{E}), we obtain \begin{align*} \int_{\Omega}-\Delta u_\varepsilon v & =\|v\|^2=\int_{\Omega}| u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon v\\ &= \int_{\Omega}| \sum (-1)^{i+1}\alpha _iP\delta _i|^{p-1+\varepsilon}(\sum (-1)^{i+1}\alpha _iP\delta _i)v\\ &\quad +p\int_{\Omega}| \sum (-1)^{i+1}\alpha _iP\delta _i |^{p-1+\varepsilon}v^2 +o(\|v\|^2). \end{align*} Hence, we have \begin{equation}\label{e:41} Q(v,v)= f(v)+ o(\|v\|^2), \end{equation} where \begin{align*} Q(v,v)=&\|v\|^2-p\int_{\Omega}| \sum (-1)^{i+1}\alpha _iP\delta _i|^{p-1+\varepsilon}v^2,\\ f(v)=&\int_{\Omega}| \sum (-1)^{i+1}\alpha _iP\delta _i|^{p-1+\varepsilon}(\sum (-1)^{i+1}\alpha _iP\delta _i)v . \end{align*} Using Remark \ref{r:24} and according to \cite{B1}, it is easy to see that \begin{equation*} Q(v,v)=\|v\|^2-p\sum_{i=1}^k\int_{\Omega}(P\delta _i)^{p-1+\varepsilon}v^2 +o(\|v\|^2) \end{equation*} is positive definite; that is, there exists $c> 0$ independent of $\varepsilon$, satisfying $ Q(v,v)\geq c \|v\|^2 $, for each $ v \in E $. Then, from \eqref{e:41} we get \begin{equation*} \|v\|^2= O(\|f(v)\|). \end{equation*} Now, using Lemma \ref{l:23}, we obtain \begin{equation} \label{e:42} \begin{aligned} f(v)&=\sum(-1)^{i+1}\int_{\Omega}( \alpha_iP\delta _i)^{p+\varepsilon}v\\ &\quad + O\Big(\sum_{i\neq j} \int_{\Omega}(\delta _i\delta _j)^{p/2}| v| +\sum_{i\neq j} \int_{\Omega}\delta _i^{p-1}\delta _j| v| (\text{if }n<12)\Big). \end{aligned} \end{equation} Using Remark \ref{r:24} and the fact that $v \in E $, we obtain \begin{align*} &\big| \int_{\Omega}P\delta _i^{p+\varepsilon}v\big| \\ &= |\int \delta_i^{p+\varepsilon}v| +O\Big(\int \delta _i^{p-1+\varepsilon}\theta_i| v|\Big) \\ & \leq c\varepsilon \int \log(1+\lambda_i^2| x-a_i|^2)\delta _i^p| v| + c| \theta_i|_{L^{\infty}}\int \delta _i^{p-1+\varepsilon}| v| \\ & \leq c\|v\|\Big(\varepsilon+ \frac{1}{(\lambda_id_i)^{n-4}} (\text{if }n<12) +\frac{1}{(\lambda_id_i)^{\frac{n+4}{2}+\varepsilon(n-4)}}(\text{if } n\geq12)\Big), \end{align*} where $\theta _i:=\theta _{a_i ,\lambda _i}:=\delta _i-P\delta _i$. For the other integrals of \eqref{e:42}, we use Holder's inequality and we obtain for $i\neq j$ \begin{align*} \int_{\Omega}(\delta _i\delta _j)^{p/2}| v| &\leq c\|v\| \Big( \int_{\Omega}(\delta _i\delta_j)^{n/(n-4)}\Big)^{(n+4)/2n}\\ &\leq c\|v\|\varepsilon_{ij}^{(n+4)/2(n-4)} (\log\varepsilon_{ij}^{-1})^{(n+4)/2n} \end{align*} and if $n<12$, we have $p-1 = 8/(n-4)>1$; therefore \begin{align}\label{e:45} \int_{\Omega}\delta _i^{p-1}\delta _j| v| \leq c\|v\| \Big( \int_{\Omega}(\delta _i\delta _j)^{n/(n-4)}\Big)^{(n-4)/n}\leq c\|v\|\varepsilon_{ij}(\log\varepsilon_{ij}^{-1})^{(n-4)/n}. \end{align} Combining \eqref{e:42}--\eqref{e:45}, the proof follows. \end{proof} Now, we need to introduce some notations before to state the crucial point in the proof of our Theorems. We denote by $G$ the Green's function defined by : $\forall x \in \Omega$ \begin{gather*} \Delta^2 G(x,.)=c_n\delta_x\quad\text{in } \Omega,\\ \Delta G(x,.)=G(x,.)=0\quad\text{on } \partial\Omega, \end{gather*} where $\delta_x$ is the Dirac mass at $x$ and $c_n=(n-4)(n-2) \omega_n$, with $\omega_n$ is the area of the unit sphere of $\mathbb{R}^n$. We denote by $H$ the regular part of $G$, that is, $$ H(x_1,x_2)=|x_1-x_2|^{4-n} -G(x_1,x_2) \quad \text{for } (x_1,x_2)\in \Omega^2\setminus \Gamma $$ with $\Gamma= \{(y,y): y\in\Omega\}$. \begin{proposition} \label{p:42} Assume that $n\geq 5$ and let $\alpha _i$, $a_i$ and $\lambda_i$ be the variables defined in \eqref{e:11} with $k=2$. We have \begin{equation} \begin{aligned} &\Big|\alpha _i c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}} -\alpha _j c_1\Big(\lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i} + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+\alpha _i \frac{n-4}{2}c_2 \varepsilon\Big| \\ &\leq c\varepsilon^2 +c \begin{cases} \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-2}}+ \varepsilon_{12}^{\frac{n}{n-4}} \log\varepsilon_{12}^{-1}+ \varepsilon_{12}^{2} (\log\varepsilon_{12}^{-1})^{\frac{2(n-4)}{n}}& \text{if } n\geq 6),\\ \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^2}+ \varepsilon_{12}^{2} (\log\varepsilon_{12}^{-1})^{2/5}\quad &\text{if } n=5, \end{cases} \end{aligned} \label{D2} \end{equation} where $i, j\in \{1,2\}$ with $i\ne j$ and $c_1$, $c_2$ are positive constants. \end{proposition} \begin{proof} Let $$ c_1=c_0^\frac{2n}{n-4}\int_{\mathbb{R}^n}\frac{dx}{(1+|x|^2)^{(n+4)/2}}, \quad c_2=\frac{n-4}{2}c_0^\frac{2n}{n-4}\int_{\mathbb{R}^n} \log(1+|x|^2)\frac{|x|^2-1}{(1+|x|^2)^{n+1}}dx. $$ It suffices to prove the proposition for $i=1$. Multiplying \eqref{ePe} by $\lambda_1\partial P \delta_1 / \partial \lambda_1$ and integrating on $\Omega$, we obtain \begin{equation}\label{e:47} \alpha_1 \int_{\Omega} \delta_1^p\lambda_1\frac{\partial P \delta_1} {\partial \lambda_1} -\alpha_2 \int_{\Omega} \delta_2^p\lambda_1\frac{\partial P \delta_2} {\partial \lambda_2} =\int_{\Omega}|u_\varepsilon|^{p-1+\varepsilon}u_\varepsilon \lambda_1\frac{\partial P \delta_1}{\partial\lambda_1}. \end{equation} Using \cite{B1}, we derive \begin{gather*} \int_{\Omega} \delta_1^p\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1} =\frac{n-4}{2}c_1\frac{H(a_1,a_1)}{\lambda_1^{n-4}}+ O\Big(\frac{\log(\lambda_1 d_1)}{(\lambda_1d_1)^{n-1}}\Big),\\ \int_{\Omega} \delta_2^p\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1} =c_1\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1} + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R, \end{gather*} where $R$ satisfies \begin{equation}\label{e:410} R = O\Big(\sum_{k=1,2} \frac{\log(\lambda_k d_k)}{(\lambda_kd_k)^{n-1}}+ \varepsilon_{12}^{\frac{n}{n-4}} \log\varepsilon_{12}^{-1}\Big). \end{equation} For the other term of \eqref{e:47}, we have \begin{equation} \label{e:411} \begin{aligned} &\int_{\Omega}|u_\varepsilon|^{p-1+\varepsilon}u_\varepsilon \lambda_1\frac{\partial P \delta_1}{\partial\lambda_1}\\ &= \int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta _2|^{p-1+\varepsilon}(\alpha_1P\delta _1-\alpha_2P\delta _2)\lambda_1\frac{\partial P \delta_1}{\partial\lambda_1} \\ &\quad +(p+\varepsilon)\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta _2|^{p-1+\varepsilon}v\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}+ O\Big(\|v\|^2+\varepsilon_{12}^{\frac{n}{n-4}} \log\varepsilon_{12}^{-1}\Big). \end{aligned} \end{equation} The above integral can be written as \begin{equation} \label{e:412} \begin{aligned} &\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta_2|^{p-1+\varepsilon} v\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}\\ &=\int_{\Omega}(\alpha_1P\delta _1)^{p-1+\varepsilon}v\lambda_1 \frac{\partial P \delta_1}{\partial \lambda_1} + O\Big(\int_{\Omega\setminus A}P\delta _2^{p-1}P\delta _1|v| +\int_{A}P\delta _1^{p-1}P\delta _2|v|\Big), \end{aligned} \end{equation} where $A = \{x : 2\alpha_2P\delta_2 \leq \alpha_1P\delta_1 \}$. Observe that, for $n \geq 12$, we have $p-1 = 8/(n-4)\leq 1$, thus \begin{align*} \int_{\Omega\setminus A}P\delta _2^{p-1}P\delta _1|v| +\int_{A}P\delta _1^{p-1}P\delta _2|v| &\leq c \int_{\Omega}|v|(\delta _1\delta _2)^{\frac{n+4}{2(n-4)}} \\ &\leq c\|v\|\varepsilon_{12}^{(n+4)/2(n-4)}(\log\varepsilon_{12}^{-1})^{(n+4)/2n}. \end{align*} For $n < 12$, we have \begin{equation} \int_{\Omega\setminus A}P\delta _2^{p-1}P\delta _1|v| +\int_{A}P\delta _1^{p-1}P\delta _2|v| \leq c \varepsilon_{12}(\log\varepsilon_{12}^{-1})^{(n-4)/n}\|v\|. \end{equation} For the other integral in \eqref{e:412}, using \cite{B1}, \cite{R2} and Remark \ref{r:24}, we obtain \begin{align*} &\int_{\Omega}P\delta_1^{p-1+\varepsilon}v\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1} \\ &= O\Big( \|v\|\Big[\varepsilon +\Big( \frac{1}{(\lambda_1d_1)^{\inf(n-4,(n+4)/2)}}(\text{if }n\neq12) +\frac{\log(\lambda_1d_1)}{(\lambda_1d_1)^4}(\text{if } n=12)\Big)\Big]\Big). \end{align*} It remains to estimate the second integral of \eqref{e:411}. We have \begin{align*} &\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta _2|^{p-1+\varepsilon}(\alpha_1P\delta _1-\alpha_2P\delta _2)\lambda_1\frac{\partial P \delta_1}{\partial\lambda_1} \\ &=\int_{\Omega}(\alpha_1P\delta _1)^{p+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}-\int_{\Omega}(\alpha_2P\delta _2)^{p+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}\\ &\quad -(p+\varepsilon)\int_{\Omega}\alpha_2P\delta _2(\alpha_1P\delta _1)^{p-1+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}+ O\Big(\varepsilon_{12}^{\frac{n}{n-4}}\log\varepsilon_{12}^{-1}\Big) . \end{align*} Now, using Remark \ref{r:24} and \cite{B1}, we have \begin{align*} \int_{\Omega}P\delta _1^{p+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1} &= \frac{n-4}{2}\Big(c_2\varepsilon +2c_1\frac{H(a_1,a_1)}{\lambda_1^{n-4}}\Big) \\ &\quad + O\Big(\varepsilon^2+ \frac{\log(\lambda_1 d_1)} {(\lambda_1d_1)^{n-1}}+\frac{1}{(\lambda_1d_1)^2}(\text{if }n=5)\Big) , \end{align*} \[ \int_{\Omega}P\delta _2^{p+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1}=c_1\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1} + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R_2, \] \begin{equation} \label{e:419} p\int_{\Omega}P\delta _2P\delta _1^{p-1+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial \lambda_1} =c_1\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1} + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R_1, \end{equation} where for $i=1,2$, \begin{align*} R_i &= O\Big(\varepsilon \varepsilon_{12}(\log\varepsilon_{12}^{-1})^{\frac{n-4}{n}}\Big) +\Big(\varepsilon_{12}^{\frac{n}{n-4}}(\log\varepsilon_{12}^{-1}) +\frac{\log(\lambda_id_i)}{(\lambda_id_i)^{n}}\text{if }n\geq8\Big)\\ &\quad +\Big(\frac{\varepsilon_{12}(\log\varepsilon_{12} ^{-1})^{\frac{n-4}{n}}}{(\lambda_id_i)^{n-4}}\text{if }n<8\Big) \end{align*} Therefore, combining \eqref{e:47}--\eqref{e:419}, and Lemma \ref{l:41}, the proof of Proposition \ref{p:42} follows. \end{proof} \section{Proof of main theorems} \begin{proof}[Proof of Theorem \ref{t:11}] Arguing by contradiction, let us suppose that the problem \eqref{ePe} has a solution $u_{\varepsilon}$ as stated in Theorem \ref{t:11}. This solution has to satisfy \eqref{e:11} and from Proposition \ref{p:42}, we have \begin{equation} \label{e:51} \begin{aligned} &c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}} -c_1\Big(\lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i} + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+\frac{n-4}{2}c_2 \varepsilon \\ & = o\Big(\varepsilon+ \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-4}}+ \varepsilon_{12}\Big), \quad \text{for } i=1,2. \end{aligned} \end{equation} Furthermore, an easy computation shows that \begin{equation}\label{e:111} \lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i}=-\frac{n-4}{2}\varepsilon_{12} \big(1-2\frac{\lambda_j}{\lambda_i}\varepsilon_{12}^{2/n-4}\big), \quad \text{for } i,j=1,2;\; j\ne i. \end{equation} Without loss of generality, we can assume that $\lambda_2 \geq \lambda_1$. We distinguish two cases and in each one, we will find a contradiction which implies our theorem. \smallskip \noindent\textbf{Case 1.} $\frac{\lambda_1\lambda_2| a_1-a_2|^2}{{\lambda_2}/{\lambda_1}}\to +\infty$. In this case, it is easy to obtain \begin{equation} \varepsilon_{12}=\frac{1}{(\lambda_1\lambda_2|a_1-a_2|^2)^{(n-4)/2}} + o(\varepsilon_{12}), \end{equation} which implies that \begin{equation}\label{e:54} \lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i}=-\frac{n-4}{2}\frac{1}{(\lambda_1\lambda_2|a_1-a_2|^2)^{(n-4)/2}} +o(\varepsilon_{12})\quad \text{for } i=1,2. \end{equation} Then from \eqref{e:51} and \eqref{e:54}, we obtain \begin{align*} &\frac{c_1}{2}\Big(\frac{H(a_1,a_1)}{\lambda_1^{n-4}} +\frac{H(a_2,a_2)}{\lambda_2^{n-4}}\Big) +\frac{c_1}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big(\frac{1}{|a_1-a_2|^{n-4}} -H(a_1,a_2)\Big)+c_2\varepsilon \\ & = o\Big(\varepsilon+ \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-4}} + \varepsilon_{12}\Big). \end{align*} Using the fact that \begin{gather*} G(a_1,a_2):=\frac{1}{|a_1-a_2|^{n-4}} -H(a_1,a_2) >0, \\ \varepsilon_{12}=O\Big(\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}+ \frac{G(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big), \end{gather*} we derive a contradiction in this case. \smallskip \noindent\textbf{Case 2.} $\frac{\lambda_1\lambda_2| a_1-a_2|^2}{{\lambda_2}/{\lambda_1}}\to c \geq 0$. In this case, we remark that $\lambda_2/\lambda_1 \to +\infty$ (since $\varepsilon_{12} \to 0$). Multiplying \eqref{e:51} by 2 for $i=2$ and adding to \eqref{e:51} for $i=1$, we obtain: \begin{equation} \label{e:57} \begin{aligned} & c_1\Big(\frac{H(a_1,a_1)}{\lambda_1^{n-4}}+2\frac{H(a_2,a_2)}{\lambda_2^{n-4}}\Big) -\frac{2c_1}{n-4}\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1}+ 2\lambda_2\frac{\partial \varepsilon_{12}}{\partial \lambda_2}\Big)\\ &- \frac{3H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}+3c_2\varepsilon \\ & = o\Big(\varepsilon+ \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-4}} + \varepsilon_{12}\Big). \end{aligned} \end{equation} Now, using \eqref{e:111} and the fact that $\lambda_2\geq \lambda_1$, an easy computation shows that \begin{equation} \label{e123} -\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1} -2\lambda_2\frac{\partial \varepsilon_{12}}{\partial \lambda_2} \geq \frac{n-4}{4} \varepsilon_{12}. \end{equation} Furthermore, since $H(a_1,a_2)\leq cd_1^{4-n}$ and $\lambda_2/\lambda_1 \to \infty$, we obtain \begin{equation}\label{e147} \frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} =o\Big( \frac{1}{(\lambda_1d_1)^{n-4}}\Big). \end{equation} Then we derive a contradiction from \eqref{e:57}, \eqref{e123} and \eqref{e147}. Our proof is thereby complete. \end{proof} \begin{proof}[Proof of Theorem \ref{t:14}] Arguing by contradiction, let us assume that problem \eqref{ePe} has solutions $(u_\varepsilon)$ as stated in Theorem \ref{t:14}. From Section 2, these solutions have to satisfy \eqref{e:11} and \eqref{e:12}. As in the proof of Proposition \ref{p:42}, we have for each $i=1,\dots, k$ \begin{equation} \label{Ei} \begin{aligned} &c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}} +c_1 \sum_{j\neq i}(-1)^{j+1}\Big(\lambda_i\frac{\partial \varepsilon_{ij}}{\partial \lambda_i} + \frac{n-4}{2}\frac{H(a_i,a_j)}{(\lambda_i\lambda_j)^{(n-4)/2}} \Big)+\frac{n-4}{2}c_2 \varepsilon \\ & = o\Big(\varepsilon+ \sum_{j=1}^k \frac{1}{(\lambda_jd_j)^{n-4}}+ \sum_{r\neq j}\varepsilon_{rj}\Big). \end{aligned} \end{equation} Observe that, if $jm$ (in the last case, we require that $(i,j)\ne (m+1,m+1)$). Observe that since $a\ne b$, it is easy to obtain that $|a_i-a_j| \geq c>0$ for each $i\leq m$ and $j\geq m+1$. Hence for $i\leq m$ and $j\geq m+1$ we have \begin{gather} \lambda_r\frac{\partial \varepsilon_{ij}}{\partial \lambda_r} = -\frac{n-4}{2}\frac{1}{(\lambda_i\lambda_j|a_i-a_j|^2)^{(n-4)/2}} +o(\varepsilon_{ij}), \quad \text{for } r=i,j,\label{wx1} \\ \varepsilon_{ij} + \frac{H(a_i,a_j)}{(\lambda_i \lambda_j)^{(n-4)/2}} =o\Big(\varepsilon_{1(m+1)}+\frac{1}{(\lambda_1d_1)^{n-4}}\Big) \quad \text{for } (i,j)\ne (1,m+1).\label{wx2} \end{gather} Now using \eqref{rome2}, \eqref{wx1} and \eqref{wx2}, we derive that \eqref{Ei'} holds for each $i\not \in \{1,m+1\}$. 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